To find the derivative of $\sin(x^3)$, we will use the chain rule.

Step-by-Step Solution:

1. Identify the outer function: The outer function is $\sin(u)$, where $u = x^3$.
2. Differentiate the outer function: The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$.
3. Identify the inner function: The inner function is $u = x^3$.
4. Differentiate the inner function: The derivative of $x^3$ with respect to $x$ is $3x^2$.
5. Apply the chain rule: Multiply the derivative of the outer function by the derivative of the inner function.

The derivative of $\sin(x^3)$ with respect to $x$ is:

$\frac{d}{dx} \left( \sin(x^3) \right) = \cos(x^3) \cdot 3x^2$

Thus, the derivative is:

$3x^2 \cdot \cos(x^3)$

Would you like more details or have any questions?

Here are some related questions:

1. What is the derivative of $\cos(x^3)$?
2. How would you find the derivative of $\sin(x^2 + 1)$?
3. What is the second derivative of $\sin(x^3)$?
4. How do you apply the chain rule for composite functions?
5. How do you find the derivative of $\sin(kx)$, where $k$ is a constant?

Tip: Always identify the outer and inner functions when using the chain rule for differentiation.